Hura Genet (1990) 86 139-146

Springer-Verlag 1990

Sex ratio of the frequencies in A: estimation and meta-analysis

F.R.Rosendaal1 2, A.H.J.T. Bröcker-Vriends1, J.C.van Houwelingen4, C.Smit2 5, I.Varekamp6, H. van Dijck5, T. P. B.M.Suurmeijer6, J.P. Vandenbroucke1, and E. Briet2 'Depaitment of Chmcal Epidemiology and 2Depaitment öl Haematology, University Hospital Leiden, ''Climcal Genetics Centre Leiden, and 4Dcpaitment of Medical Statistics, State University Leiden, Leiden, The Netheilands 5Dutch Haemophilia Society (NVHP), Amsterdam, The Netherlands 6Department of Medical Sociology, State University Groningen, Groningen, The Netheilands

Received Febmaiy 2, 1990 / Revised June 18, 1990

Summary. A hereditary disease with excess mortality such tion despite excess mortality. may occur either äs haemophilia is maintained in the population by the in females or in males and give rise to male patients or occurrence of new cases, i.e. mutations. In haemophilia, female carriers. Patients produce additional carriers mutations may arise in female or male ancestors of a when they procreate, whereas carriers may give birth to 'new' patient. The ratio of the mutation frequencies in patients or may silently pass the on to their daugh- males over females determines the prior risk of carrier- ters. Therefore, an isolated patient, i.e. a patient with- ship of the mother of an isolated patient. An estimate of out prior family history, may be the result of a mutation this prior risk is required for the application of Bayes' in his mother and may then be called a true sporadic case, theorem to probability calculations in carriership testing. or he may be the son of a carrier. In the latter case, the We have developed a method to estimate the sex ratio of mutation occurred in the grandparental or a further re- the mutation frequencies; it does not depend on the as- moved generation. sumption of genetic equilibrium, nor require an estimate The probability that an isolated patient is a truly of the reproductive fitness of haemophilia patients and sporadic case is the complement of his mother's risk of carriers. Information from 462 patients with severe or being a carrier, and depends on the relative mutation moderately severe was gathered by postal frequencies in both sexes. If mutations occur predomm- questionnaires in a survey that included practically all antly in males, it becomes likely that the mutated gene Dutch haemophiliacs. Pedigree analysis was performed carried by an isolated patient originated in one of the for the 189 patients of these 462, who were the first male ancestors of the patient's mother, who then would haemophiliacs in their family. By the maximum likeli- be a carrier. On the other hand, if mutations occur pre- hood method, the ratio of the mutation frequencies in dominantly in females, the mutated gene may well have males and females was estimated at 2.1, with a 95% con- originated in the mother of the isolated patient. Therefore, fidence interval of 0.7-6.7. In addition, we performed a the sex ratio of the mutation frequencies determines the meta-analysis of all published studies on the sex ratio of risk of carriership of the mother of an isolated patient. the mutation frequencies. When the results of six studies were pooled, it was estimated that mutations originated The indirect method 3.1 times äs often in males äs in females. The 95% confi- dence interval was 1.9-4.9. This implies that 80% of In his classic article of 1935, Haldane presented the indi- mothers of an isolated patient are expected to be haemo- rect method to estimate the mutation rate of a sex-linked philia carriers. disease under the assumption of a balance of selection and mutation. He showed that, in genetic equilibrium, the proportion of sporadic cases to all cases can be used to estimate the sex ratio of the mutation frequencies of Introduction an X-linked disease. When patients' reproductive fitness ((/A) is nil, such äs in Duchenne's muscular dystrophy, Only by the occurrence of new cases via mutation is a and the mutation frequencies in males (v) and females disease such äs haemophilia maintained in the popula- (μ) are equal, this proportion χ is one third of the total number of patients. According to Haldane's equation, Offprmt requests to F. R. Rosendaal, Depaitment of Chmcal Epi- χ = μ/(2μ + v), χ will be lower than one third when mu- demiology, Building l, CO-P-46, Umveisity Hospital Leiden, tations occur more frequently in males than in females. P.O.Box 9600, NL-2300 RC Leiden, The Netheilands This has been called the 'deficit of sporadic cases'. 140

When fitness (f^) increases, the distribution of giving rise to individuals who bear a mosaic of cells with sporadic and hereditary cases will shift towards heredit- and without the mutated gene. These individuals may ary cases and χ will decrease. This can be corrected for, pass on the affected gene to more than one of their off- i.e. Xf= (I-/A)*O· A method to adjust for a decreased fit- spring. Mosaicism may exist in the gonads only (germ ness of carriers has also been proposed (Holloway and line mosaicism) or may be extended to all body tissues Smith 1973). The usual methods in segregation analysis (somatic mosaicism) of males or females (Bakker et al. use pedigree Information to estimate the proportion of 1987; Higuchi et al. 1988; Maddalena et al. 1988; Hall sporadic cases to all cases (Morton and Chung 1959). 1988; Bröcker-Vriends et al. 1990). With respect to mu- These methods, however, have serious drawbacks. First, tation rates, these two types of mosaicism are not differ- they depend on the assumption of genetic equilibrium. ent; both result from mutations during an individual's Secondly, they require a good estimate of patients' fit- development from a fertilized zygote to the gamete-pro- ness. Thirdly, they are sensitive to incomplete ascertain- ducing adult. The population mutation rate is defined äs ment and ascertainment bias. the proportion of X-chromosomes per generation chang- Since the beginning of this Century, the life expec- ing from the unaffected state to the affected state, and tancy of patients with severe haemophilia has increased not äs the mutation rate per cell division (Haldane 1935; steadüy, spectacularly so since the 1960s, from a mere 11 Murphy et al. 1974). Generally, the mutation rate can be years in 1900 to 65 years most recently (Ikkala et al. viewed äs the probability that an individual who did not 1982; Larsson 1985; Rosendaal et al. 1989). The effec- inherit a haemophilia gene passes on such a gene to his tive fitness that has led to the present distribution of or her offspring. The phenomenon of mosaicism does sporadic and hereditary cases is thus difficult to assess, not alter this definition of the mutation rate or its sex whereas the assumption of a steady state is completely ratio, both of which are the results of several possible unwarranted. Segregation analysis is usually performed events including mosaicism. Theoretically, it implies that under incomplete ascertainment (Morton 1959), which if the sex ratio differs from unity, this may not only be can only be corrected for by estimation of the ascertain- caused by a higher rate of mutation per cell division in ment probability. Hereditary cases are often more likely one of the sexes, or by a larger number of cell divisions to be included in a study than isolated cases; this may in one of the sexes at an equal mutation rate per divi- lead to biased results (ascertainment bias). sion, but also by mutations generally occurring at an ear- lier stage of development in one of the sexes. Moreover, Equilibrium independent method it is no longer justified to discuss mutation rates in sperm and ova: this should be the mutation rates in males and We have developed an approach to estimate the sex ratio females. of the mutation frequencies; our method avoids these difficulties. In the absence of equilibrium, the increased fertility of patients that causes the imbalance will only Meta-analysis lead to more hereditary cases. The mutation rates will For the two most common X-linked recessive diseases, still be constants. The number of initially isolated cases, Duchenne muscular dystrophy and haemophilia, several i.e. sons of mothers without a prior family history of studies using various methods have been carried out, haemophilia, will therefore be dependent only on the with different results. In a recent review of 17 studies of mutation rates in males and females. A number of these Duchenne muscular dystrophy (Moser 1984), it was con- will be sporadic patients, i.e. sons of non-carrier women; cluded that there was no evidence for a difference in the their number will be dependent only on the female mu- mutation rates in males and females. For the less com- tation rate. Therefore, when the analysis is limited to in- mon Lesch-Nyhan disease, only a few studies have been itially isolated cases, the assumption of a steady state is performed; here, the conclusion tends to be that the not needed, nor is the fitness of patients relevant to the male mutation rate exceeds the female rate (Francke et calculation. This restriction to initially isolated cases also al. 1976, 1977; Winter 1980). limits the possibility of ascertainment bias, äs has been In haemophilia, the results are not in agreement. As suggested before (Winter 1980). In our calculation, the early äs 1947, Haldane calculated a higher mutation fre- distribution of healthy and affected sibs of initially iso- quency for haemophilia in males than in females. Based lated cases is used to estimate the sex ratio of the muta- on a study of Andreassen of 63 pedigrees (Andreassen tion frequencies. We have set up a model in which the 1943), he suggested a ratio of v to μ of more than 10:1. probability that a woman has more than one affected Several studies have been conducted since, some of which son, given that she has one affected son, is expressed äs have confirmed this finding (Hermann 1966; Biggs and a function of the sex ratio of the mutation frequencies, Rizza 1976; Winter et al. 1983), whereas others did not which is then estimated by the maximum likelihood (Kosower et al. 1962; Barra'i et al. 1985). An important method. By this method, it is possible to estimate the sex reason for the seemingly conflicting results of these ratio of the mutation rates of an X-linked disorder in the studies may be the lack of statistical power to detect a absence of genetic equilibrium. difference in the mutation rates (Karel et al. 1986). This power can be increased by pooling the results of all avail- Mosaicism able studies in a so-called meta-analysis (Sacks et al. Recently, it has been shown that a mutation may cause a 1987). The aim of a meta-analysis is to obtain a more re- clone of cells with the affected gene to develop, thereby liable and precise estimate than was possible in the indi- 141 vidual studies We therefore mcluded the result of our wheie Θ = μΐ Ω (see Appendix for denvation of equations) Smce analysis m a meta-analysis of all available studies on the a carner woman without family history will either be the result of a mutation or the daughter of a carner (without family history), sex ratio of the mutation frequencies m haemophiha A Ω = v + μ + 1/2Ω, so that Ω = 2 v + 2μ Smce Θ = μ/Ω it follows that the ratio ot the mutation frequencies is Practical importance

Smce the sex ratio of the mutation frequencies deter- 2Θ mmes the nsk of carnership of the mother and other female relatives of an isolated patient, knowledge of this The probabihty that the mother of an isolated patient is a carner (C) is !/2Ω/(ί/2Ω + μ), which is ratio is of practical importance m climcal genetics. v + μ l P(C\ mothei of isolated patient) = Eq III ν+2μ 1 + 2Θ Methode An estimate of Θ will yield the latio of male to female muta- Data collection tion frequencies and the nsk of carnership of the mother of the iso The data were obtamed by postal questionnaires These were sent lated patient to 1162 haemophiha patients, who were identified by an extensive The distribution of sibships with one affected son and more seaich covermg most Dutch hospitals, the Dutch Haemophiha than one affected son served to produce a maximum hkelhood esti- Society and the files of our earher surveys of 1972 and 1978 Of the mate of Θ and its Standard error usmg Eq I (see Appendix for 81% (947 patients) returmng the form, 935 weie smtable for analy maximum hkehhood method) The standaid error for Θ was esti- sis The analysis was restncted to haemophiha A and to patients mated via the second derivative of the log-hkehhood function with seveie or moderately severe haemophiha only, i e with less Confidence mteivals foi Eqs II and III were calculated after log than 5% (5IU/dl) coagulant activity factor level, assayed m the transformation, i e via log(0) for the probabihty of cainership of Standard fashion (Veitkamp et al 1968) Of these 466 patients, a the mother of an isolated patient and via log (ν/μ) for the sex ratio further 4 had to be excluded 2 because they were part of homo- zygous twms (one mdividual of each pair of twms was mcluded), Meta-analysis and another 2 smce they had been adopted and all pedigree Infor- mation was missmg Consequently, the data presented here refer We searched for studies m which an estimate of the sex ratio of the to 462 patients mutation frequencies was present This also mcluded studies m The questionnaire was standardized and contamed questions which analyses were performed on data collected by other work- conceinmg the numbei of affected and unaffected sibhngs and ers The studies were gathered by cross referencmg and an on-hne whether the patient was the fnst haemophihac m the family, it also search (database Medlme, Cologne, FRG) mcluded questions pertaming to demographic and general medical We used the natuial logarithm of the sex ratio ν/μ from each variables An extensive report of the fmdmgs has been pubhshed study and the Standard error of log(ν/μ) Imtially, the estimates (äs elsewhere (Smit et al 1989) log ratio) were pooled in the classical precision weighted manner, In order to estimate the ratio of male to female mutation fre- with the mverse squared Standard errors äs weights [w = l/(s e )2], quencies and the probabihty of carnership of the mother of an iso the Standard error of the weighted average was approximated äs lated patient, only Information about the piogeny of those women the mverse of the square root of the sum of the weights (s e = l/ who had no family history of haemophiha befoie the birth of their ]/(ΣΪν)) This method depends on a stnngent assumption of homo affected son was considered We used Information about all other geneity among the studies (Greenland 1987) sons of these mothers m our calculations, and not only of the sons We mvestigated whether the studies were homogeneous, i e born after the haemophihc child, this resembles the subsequent whether the differences m the reported outcomes could be attn sibs method (Lane et al 1983) Our appioach obviously mcreases buted solely to chance Variation aiound the 'true' parameter 2 the statistical power, but would mtroducc an eiroi if women tended Homogeneity was tested by calculatmg G = Σ w,(k, — ke) m which to hmit the size of theu famihes after the buth of the haemophihc w, is the weight given to each study that was used to obtam the child This can be understood by lookmg at the extreme if no weighted average, /c, the estimate of 1ο§(ν/μ) m each study, and ke women at all had additional children after the birth of the affected the pooled estimate G follows a x2-distnbution with n-1 degrees of child, we would ncver see sibships with more than one affected freedom for n studies child, and we would grossly under-estimate the sex ratio of the mu Smce the studies proved heteiogeneous, we apphed two other tation frequencies Therefore, we proceeded with the analysis only methods of obtammg a summary estimate First, the Odd man out' after it was established that family size restnction was absent m oui method, a graphical approach recently presented by Walker et al pedigrees, by comparmg observed and expected birth lanks (see (1988) With this method, heterogeneity will, accordmg to Walker Results) et al , lead to mtervals remammg wide, or to the emeigence of two mtervals Secondly, we pooled the study results by treatmg the Estimation method pomt estimates äs single observations, i e without Variation esti- mates The summary estimate is simply the unweighted average Let v be the mutation frequency m males, μ in females and let Ω (of the log ratio) of the pomt estimates, wheieas the Standard erioi be the probabihty of carnership of a woman without a family histo- is calculated m the Standard fashion for n (numbei of studies) ob ly of haemophiha Let i be the number of affected sons and s the servations The conservative method allows for heterogeneity total number of sons The Status of the other sons of a woman without a pnor family history, boin before or after the haemo phihc son (the imtially isolated patient) deteimmes the piobabihty of cainership When we assume that the recurrence nsk for non- Results carneis is neghgible, all subsequent births of affected sons will m- ciease the nsk of carriership to umty, whereas the occurrence of Estimation of the sex ratio unaffected sons will decrease it The probabihty of havmg more than one affected son, given one affected, is Two hundred and seventy-three (59%) of the patients reported that they had relatives with haemophiha at the P(r>l\r>U)= Eq I üme of their birth, and may be considered hereditary 4 cases One hundred and eighty-nme patients (41%) re- V ' ' 142 Table 1. Affected male sibs by male sibshipsize for initmlly isolated reported (Haldane 1947, Kosower et al 1962, Biggs and patients (n = : Rizza 1976, Ananthaknshnan and D'Souza 1979, Winter No Sibshipsize (index patient mcluded) (s) et al 1983, Barrai et al 1985) In two studies, the analy- affected sis had been performed on data collected previously by l (r) other workers (Andreassen 1943 in Haldane 1947, Birch 1937, Hoogvhet 1942, Andreassen 1943, Fomo 1954 m r = 1 74 56 18 5 1 1 2 0 Kosower et al 1962) In four studies, the results of car- r>l - 14 6 3 2 4 2 1 ner detection tests in the mother of isolated patients had r = 2 - 14 6 2 1 1 0 0 been used to determme the fraction of sporadic patients r = 3 0 0 1 2 2 1 (Haldane 1947, Biggs and Rizza 1976, Ananthaknshnan r = 4 - 1 0 1 0 0 and D'Souza 1979, Winter et al 1983), whereas m the other studies, only pedigree Information had been used (Kosower et al 1962, Barrai et al 1985) The number of ported that they had been the first haemophihac m the patients ranged from 21 isolated cases to 669 sibships family (mitially isolated) The reported estimates of the sex ratio ν/μ vaned from To determme whether women had hmited family size l 2-29 3 after the birth of the haemophihc chüd, we compared In only two studies, mcludmg ours, had the analysis observed and expected birth ranks of the mitially iso- been hmited to more severe forms of haemophilia (Win- lated patients The observed birth rank of the 189 pa- ter et al 1983) m order to reduce the possibility of ascer- tients, (Es)/« for n sibships, was 2 15 The expected birth tamment bias In all studies except ours, a method was rank, calculated äs {L(s + 1)12}In, was 2 16 If a tendency used that was dependent on genetic equihbrium The es- to limit family size had been present, the observed birth timates of patients' fitness used in these studies vaned rank would have exceeded the expected birth rank from 0 28-0 70 Smce this appeared not to be the case, Information on all We excluded Haldane's paper (1947) from the final sibs of the mitially isolated patient, mcludmg those born analysis, since the time, used by Andreas- before him, could be used in the analysis sen, is not generally prolonged m haemophilia carners The distnbution of healthy and affected male sibs of (Merskey and MacFarlane 1951) Moreover, Andreas- the mitially isolated patients is shown in Table l The 74 sen's pedigrees were also mcluded m the analysis of patients who had no brothers were not mcluded m the Kosower et al (1962) Although the data of Kosower et maximum hkehhood estimate, since they contnbuted no al (1962) also compnsed patients, we de- Information to the genetic Status of their mother The cided to include their results, since these patients are remammg 115 patients had a total of 210 brothers, 42 only a small fraction of all haemophilia patients (Larsson affected and 168 unaffected We used Eq I to obtain et al 1982, Smit et al 1989) a maximum hkehhood estimate for Θ of 0 16046 (s e = 0 06407) The ratio of male to female mutation frequencies 100-] can now be calculated by Substitution from Eq II, ν/μ = (l - 2Θ)/2Θ = 2 12 This imphes that mutations 41 5 ongmate more than twice äs often m males äs in females The 95% confidence mterval, which was made log-sym- metncal around ν/μ, was 0 7-6 7 Since the probability of carnership for the mother of 10: 67 an isolated patient is (v+ μ)/(ν+ 2μ) = 1/(1 + 2Θ), this 56 42 probability has a theoretical ränge from 50%-100% From Eq III, we find that 76% of mothers of an isolated Φ (Λ 22 ~- patient will be carners, with a 95% confidence mterval, 1 9 via log (Θ), of59%-87%1

07 Meta-analysis We found six studies, m addition to ours, m which an es- timate of the sex ratio of the mutation frequencies was

1 List of Symbols 123456 ABC v mutation rate m males Fig. 1. Meta-analysis of six studies The pomt estimates are shown μ mutation rate m females of six studies of the sex ratio of the mutation frequencies m males /4 fitness of patients and females, with the 95% confidence mterval Both the summary χ proportion sporadic patients on all patients results by the precision weighted method (A), and the Odd man ; number of affected males m a sibship out procedure (B) are shown, together with the estimate calcu i total number of males in a sibship lated by treatmg the study results äs smgle obseivations (C) Ω probability of carnership of a woman without a family history Studies ] Kosower et al (1962), 2 Biggs and Rizza (1976), 3 Anan of haemophilia thaknshnan and D'Souza (1979), 4 Winter et al (1983), 5 Banai et Θ μ/Ω al (1985), 6 Rosendaal et al (this study) 143 We modified the results reported by Biggs and Rizza it is inconceivable that patients with severe or moder- (1976). Among 41 mothers of isolated patients, from a ately severe haemophilia would not have been diagnosed. total of 86 mothers, they found 37 carriers by factor VIII Indeed, we noticed that many patients were known in measurements. Subsequently, in the families of the four several hospitals. The non-response was 19%, but is un- women considered homozygous, carriership testing was likely to be related to genetic Status. We conclude there- performed in other female relatives; this led to two more fore that ascertainment of patients was unbiased. women being considered carriers. We ignored this bi- Our method to estimate the sex ratio of mutation fre- ased search, which reduced the estimate from 29.3 to quencies is independent of patients' fitness/4, and is in- 13.5. The Standard errors reported by Kosower et al. sensitive to distortion of equilibrium. It does rely on the (1962) and Barrai et al. (1985) were modified to obtain patients' or their mothers' recollection that no cases oc- log-symmetrical confidence intervals. When only a point curred previously in a family. Haldane has shown that estimate was reported (Biggs and Rizza 1976; Ananthak- the mean life of a haemophilia gene in the population rishnan and D'Souza 1979), we calculated the Standard is 3/(1-/)-3μ/(2μ + ν), which is, for fA = 0.25-0.40, error for the proportion of mothers reported to be car- three to four generations (Haldane 1935). Since a fair re- riers. collection up to three generations may be expected, and The summary estimate of the sex ratio äs calculated since a gene will be passed on silently from female to in the precision-weighted method was 3.1, with a 95% female over several generations only rarely, we feel that confidence interval of 1.9-4.9. The confidence interval this does not introduce an error of importance. obtained by the Odd man out' method was almost identi- cal: 2.2-5.6 (Fig. 1). Since the studies were heterogene- Meta-analysis ous (G = 13.3, P< 0.025), we calculated a straight aver- age and a Standard error between the six Observations'; The studies that we used to calculate a summary esti- this resulted in a summary estimate of 3.1 (95% confi- mate were not homogeneous, i.e. the results showed dence interval 1.4-7.1). more Variation around the summary estimate than could be explained by chance alone. Nevertheless, we con- clude that the mutation frequency for haemophilia A in Discussion males exceeds that in females, since even the most con- servatively calculated summary confidence interval did We have estimated the sex ratio of mutation frequencies not include one, whereas in none of the six studies was using an equilibrium independent method, and have sub- an estimate of the sex ratio of less than one found. sequently performed a meta-analysis of all available This conclusion is in accordance with studies in studies on this subject. We conclude that mutations that which, instead of the calculation of an estimate of the sex cause haemophilia A occur more often in males than in ratio, a hypothesis was tested. Bitter (1964) could not re- females, at a ratio of about 3:1. This implies that about ject the hypothesis that all mothers of haemophilia pa- 80% of mothers of an isolated patient carry the gene for tients were carriers in a survey in the Hamburg area. Vo- haemophilia. Therefore, carrier testing by factor VIII discriminant analysis and DNA analysis remains impor- tant in these women. 1 0- Estimation of the sex ratio 09- The results of a segregation analysis may be flawed be- 0 ί cause of ascertainment bias, which occurs when the 07 - probability of being a study subject is dependent upon _ 06 - genetic Status. Usually, the bias results from an over- representation of hereditary cases, because these are οΓ 05 - more readily identified äs patients, since they are regis- 04 - tered äs f amilies in hospitals or genetic centres. When it is the aim of a study to estimate the proportion of spo- 03- radic cases to all patients, it is obvious that this might 02 - lead to erroneous conclusions. The problem of ascertain- ment bias was limited in the method presented in this Ο 1 - paper, because only initially isolated patients and their 1/5 5 male siblings were included. Furthermore, patients with 1/501/401/301/201/10 1 10 20 30 40 50 mild haemophilia were excluded, since their disease may remain undetected for many years, whereas detection of these patients will be much more likely for familial cases. Fig. 2. The nsk of carneiship of the mother of an isolated patient We carried out an extensive search that included most with haemophilia A is shown äs a function of s (total number of sons, of whom one is the haemophilia patient) and k, the ratio of Dutch hospitals, with the help of the patients' and hae- the mutation fiequencies m men and women (k = ν/μ) The func- mophilia physicians' organization. We are convinced tion can be denved by calculatmg the odds of carriership, with a that we reached virtually all Dutch haemophiliacs, since pnor probability of carneiship of 2 v + 2 μ 144 gel (1965), who used data from Sjolin (that included An- occur at each cell division, this will only rarely take place dreassen's pedigrees), Fonio, Ikkala and Bitter obtained at the very few first postzygotic divisions. Therefore, in the same result. He also reported that the data were very mosaic women, only a small fraction of cells will eventu- clearly in disagreement with a hypothesis of equal muta- ally bear the affected gene. Although the recurrence risk tion rates in males and females (μ— ν). will be much higher than the mutation rate in these Several studies have been conducted to determine cases, it will still be negligibly small. A sizeable effect of the number of carriers among mothers of isolated pa- mosaicism will only be present when the mutation occurs tients, with very different results. Ratnoff and Jones in the earliest phase of embryonic development or gona- (1977) classified 85% of 39 mothers of isolated patients dal development. This seems generally unlikely, unless äs carriers by factor VIII level discriminant analysis, the human organism is more sensitive to mutations in the whereas Ekert (1977), using the same method, found early developmental phase than in the later stages of de- only 47% (of 32 women). These results, however, should velopment. Moreover, äs pointed out by Hall (1988), the not be compared and give no Information on the sex ratio frequency of somatic mosaicism might be different for of the mutation frequencies. The risk of carriership of different mutations. Since mosaicism affects the reliabil- the mother of an isolated patient is dependent on the sex ity of carrier testing in clinical genetics, an estimate of its ratio of the mutation frequencies (the prior probability) importance for different disorders is needed. A possible and on the number of (unaffected) sons, i.e. in a popula- approach would be to investigate the distribution of af- tion with a high average number of children, mothers of fected and unaffected sons in sibships with two or more isolated patients will rarely be carriers. Figure 2 shows affected children, to observe a possible departure from a the risk of mothers of isolated patients äs a function of segregation frequency of 0.50. the number of children and the sex ratio of the mutation frequencies. Conclusion Recently, the results of carrier testing by restriction fragment length polymorphisms (RFLP) in 17 families One mechanism that has been offered äs an explanation with an isolated patient were reported (Bernardi et al. for a possible difference in mutation frequencies in 1987). In 8 families, carriership of the maternal grand- males and females is that female germ cells are produced mother could be excluded. In all of these families, the by a limited number of mitotic cell divisions occurring mothers were carriers: 6 had inherited the X-chromo- before birth and only two divisions (the meiotic divi- some carrying the mutant gene from their father, 2 from sions) in the adult woman. In the male, however, numer- their mother. This study, although it may be questioned ous mitotic divisions occur throughout life, and so the since mutations in the grandpaternal X-chromosome are number of divisions is dependent on age (Thompson and more readily identified than in the grandmaternal, is Thompson 1966; Vogel 1977). compatible with a sex ratio of 3. In one study (Hermann 1966), it was reported that the maternal grandparents of haemophiliacs were older Mosaicism than expected, whereas mothers of isolated patients had a higher birth-rank than expected. This result, obtained In 1935, Haldane hinted at the possibility of somatic by a completely different approach, gives further evi- mosaicism for haemophilia when he wrote that the mu- dence that the mutation rate is higher in male gametes tation in the mother of two patients with haemophilia possibly because more cell divisions occur in the de- might have occurred 'during her early embryonic life'. In velopment of male germ cells than of female gametes. his paper of 1947, he offered äs a general possibility for the occurrence of new haemophilia : 'the gene for Acknowledgements We gratefully acknowledge the kind coopera- haemophilia may have arisen by mutation in part of her tion of the patients and the haemophilia treatment centres We also thank Dr.L.P.Ten Kate for promptmg us to Start this analy- body, including the ovaries in whole or part' (Haldane sis. The study was made possible by fmancial support from Het 1947). Praeventiefonds (28-1099) and De Stichtag Haemophilia. Although the possibility of mosaicism for haemo- philia was considered long before it was actually demon- strated, it has become customary in the practice of clini- References cal genetics to discount it. This is also reflected in the guidelines for recognition of obligatory carriers, äs all Akhmeteli MA, Aledort LM, Alexamants S, Bulanov AL, Eiston mothers of two or more haemophilic sons are included RC, Gunter EK, Goussev A, Graham JB, Hermans J, Larneu (Akhmeteli et al. 1977). MJ, Lothe F, McLaren AD, Manucci PM, Prentice CRM, Veit- kamp JJ (1977) Methode foi the detection of haemophilia cai- Theoretically, female mosaicism would affect our ners: a memorandum. Bull WHO 55 675-702 method of estimation, since in our model, we assume Ananthaknshnan R, D'Souza S (1979) Some aspects of the occur- mothers of two or more haemophilic sons to be carriers. rence of new mutations m haemophilia. Hum Hered 29 90-94 This would lead to an under-estimate of the female mu- Andreassen M (1943) Haemophilia m Denmark Opera ex domo tation rate, i.e. an over-estimate of the sex ratio. It has biologiae hereditanae humanae (in Danish). Umversitatis Haf- been shown that when a constant mutation rate per cell mensis (Copenh) 6:1-168 Bakkei E, Broeckhoven C van, Bonten EJ, Vooien MJ van de, division is assumed, mosaicism has very little effect on Veenema H, Hui W van, Ommen GJB van, Vandenbeighe A, the posterior probabilities (Murphy et al. 1974). This can Pearson PL (1987) Geimlme mosaicism and Duchenne muscu- intuitively be understood äs follows: when mutations can lar dystrophy mutations Nature 329 554-556 145 Banai I, Cann HM, Cavalh Sforza LL, Baibujam G, Nicola P de Meiskey C MacFailane RG (1951) The female canier of hemo- (1985) Segiegation analysis of hemophilia A and B Am J Hum philia a clmical and laboiatory study Lancet I 487-490 Genet 37 680-699 Moiton NE (1959) Genetic tests undei mcomplete ascertainment Bemaidi F, Marchetti G, Beitagnolo V, Faggioh L Vohma S, Pat Am J Hum Genet 11 1-16 lacchmi P, Baitolai S, Vannim F, Fellom L Rossi L, Pamcucci Morton NE, Chung CS (1959) Foimal genetics of muscular dys F Conconi F (1987) RFLP analysis m families with sporadic trophy Am J Hum Genet 11 360-379 hemophilia A estimate of the mutation latio m male and female Mosel H (1984) Review of studies on the pioportion and ongm of gametes Hum Genet 76 253-256 new mutants in Duchenne musculai dystiophy In Kate LP ten, Biggs R Rizza CR (1976) The spoiadic case of haemophiha A Lan Peaison PL Stadhoudeis AM (eds) Research mto the origm cetll 431-433 and tieatment of musculai dystiophy Excelpta Medica, Am Buch CL (1937) Hemophilia, clmical and genetic aspects Illinois sterdam, pp 41-52 Monogi Med Sei l 1-151 Muiphy EA Ciamei DW, Kiyscio RJ, Brown CC Pierce ER Bittei K (1964) Eihebungen zui Bestimmung dei Mutationsiate fui (1974) Gonadal mosaicism and genetic counselmg for X hnked Hamophilie A und B in Hambuig Z Menschl Veieibungs recessive lethals Am J Hum Genet 26 207-222 Konstitutionsl 37 251-268 Ratnoff ÖD Jones PK (1977) The laboiatory diagnosis of the car- Biockei Vuends AHJT, Briet E, Dreesen JCFM, Bakkei E, nei state for classic hemophilia Ann Intern Med 86 521-528 Reitsma PH, Pannekoek H, Kamp JJP van de Peaison PL Rosendaal FR, Varekamp I, Smit C, Biocker Vuends A, Van Dijck (1990) Somatic ongm of inhented haemoplulia A Hum Genet H Vandenbroucke JP, Hermans J, Suuimeijei TPBM, Briet E 85 288-292 (1989) Moitality and causes of death m Dutch haemophihacs, Ekert H (1977) Camei studies m 'Simplex families" Thiomb 1973-1986 Bi J Haematol 71 71-76 Haemost38 721-723 Sacks HS, Bernei J, Reitman D Ancona-Berk VA, Chalmers TC Fomo A (1954) Die eiblichen und die spoiadischen Bluteistamme (1987) Meta analyses of landomized contiolled tnals N Engl J m dei Schweiz Bull Schweiz Akad Med Wiss 10 303-421 Med 316 450-455 Francke U, Felsenstem J, Gaitlei SM, Migeon BR Dancis J Seeg- Smit C, Rosendaal FR, Vaiekamp I, Biockei Vuends A, Van Dijck millei JE, Bakay F, Nyhan WL (1976) The occmience of new H, Suuimeijei TPBM, Briet E (1989) Physical condition, longe mutants in the X hnked lecessive Lesch-Nyhan disease Am J vity and social perfoimance of Dutch haemophihacs 1972-1985 Hum Genet 28 123-127 BrMedJ298 235-238 Fiancke U, Felsenstem J, Gartlei SM, Nyhan WL, Seegmillei JE Thompson JS, Thompson MW (1966) Genetics m medicme Saun- (1977) Answei to cuticism of Moiton and Lalouel Am J Hum ders, 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te, Kate LP ten (1986) On the powei to de- Derivation of equations I, II and III tect diffeiences between male and female mutation lates foi Duchenne musculai dystiophy, usmg classical segiegation anal Let s be the total number of sons m a sibship, r the ysis and restriction fiagment length polymoiphisms Am J Hum number of affected sons, Ω the probabihty of carnership Genet 38 827-840 of a woman without a family history of haemophiha and Kosowei N, Chustiansen R, Moiton NE (1962) Spoiadic cases of μ the female mutation rate In the general form, the hemophilia and the question of a possible sex difference m mu- probabihty of r affected sons can be wutten äs tation lates Am J Hum Genet 14 159-171 4 Lane RJM, Robinow M, Roses AD (1983) The genetic Status of P(r) = (?) [Ω(ΐ/2) + (l - Ω) μ'(l - μ)*-'] mothers of isolated cases of Duchenne musculai dystiophy J When we assume that μ and Ω are of the same small Med Genet 20 1-11 Laisson SA (1985) Life expectancy of Swedish haemophihacs, order of magmtude, terms mvolvmg μΩ., μ , äs denoted 1831-1980 Br J Haematol 59 593-602 by , will be neghgibly small It follows that Laisson SA, Nilsson IM, Blomback M (1982) Cunent Status of P( = 0) = (ΐ/ )Ώ + (l - Ω) (l - μ)5 Swedish hemophiliacs Acta Med Scand 212 195-200 r 2 Maddalena A, Sosnoski DM, Beny GT, Nussbaum RL (1988) = i - Ω [i - (i/2y] - ä/i + Mosaicism foi an mtragemc deletion m a boy with mild 01 and mthme transcaibamylase deficiency N Engl J Med 15 999- 1003 p (r = l) = 146

These can be combined into: The probability of the mother of an isolated patient P(r>l) l - P(r = 0) - P(r = 1) being a carrier (C) is: P(r>l ~P(r>l) ^ P (r = 0) P(C\ mother of isolated patient) =

_ v + μ _ l Ω [l - (1/2)'] + fjs + .... Eq. III. ν+2μ+.... 1+20 By substituting Θ for μ/Ω, dividing numerator and de- nominator by Ω, and leaving out the negligibly small terms, we obtain Maximum likelihood eslimate (MLE) P(r> l r> 1) is a function of s and 0: p (s,®). n l-(1/2)* +0s MLE 0 of 0 maximizes: L(0) = Π/?(Χ0)^'[1 ~p(s,®)]Y* Ω can be approximated into an expression of the male mutation rate v and the female mutation rate μ; this is in which X^ is the number of families with r > l and Y, most easily derived from its complement l — Ω, by con- the number of families ^with r = 1. ditioning on the Status of the maternal grandmother: The Standard error of 0 is obtained in the usual way via l — Ω = P(non-carrier) the second derivative of the log-likelihood function, i.e. -l = Ωΐ/2 (l - v)(l - μ) + (l - Ω)(1 - v)(l - μ) s.e.2 = , attheMLE(0). = (l - ι/2Ω)(1 - v)(l - μ), and therefore: log [L(0)] d& 'l-i/2(l- v)(l-μ) 0 — 0 Approximately, has the Standard normal distri- s.e. (0) 1/2 bution. This does not imply that mean (0) = 0, nor that 2 Equations II and III can now be derived by Substitution: var (0) = s.e (0), but it does imply that , 2ν , , , 2ν+2μ . Ω , ν/μ = — + l - l = ——- - l = -— l = 2μ 2μ 2μ s.e. (0) TJ —)/72π) _ _L _1-2Θ The normal approximation can be used to construct (ap- J^ — Eq. II. proximate) confidence intervals for 0.